Figure 1:
This figure shows a crosssection of a coordinate shape (the thick curve) which is not a Strahlkörper about the local coordinate origin shown (the large dot). The dashed line shows a ray from the local coordinate origin, which intersects the surface in more than one point. 

Figure 2:
This figure shows part of a simulation of the spherically symmetric collapse of a model stellar core (a polytrope) to a black hole. The event horizon (shown by the dashed line) was computed using the “integrate null geodesics forwards” algorithm described in Section 5.1; solid lines show outgoing null geodesics. The apparent horizon (the boundary of the trapped region, shown shaded) was computed using the zerofinding algorithm discussed in Section 8.1. The dotted lines show the world lines of Lagrangian matter tracers and are labeled by the fraction of baryons interior to them. Figure reprinted with permission from [142]. © 1980 by the American Astronomical Society. 

Figure 3:
This figure shows a number of light cones and futurepointing outgoing null geodesics in a neighborhood of the event horizon in Schwarzschild spacetime, plotted in ingoing Eddington–Finkelstein coordinates . (These coordinates are defined by the conditions that is an ingoing null coordinate, while is an areal radial coordinate.) Note that for clarity the horizontal scale is expanded relative to the vertical scale, so the light cones open by more than . All the geodesics start out close together near the event horizon; they diverge away from each other exponentially in time (here with an folding time of near the horizon). Equivalently, they converge towards each other if integrated backwards in time (downwards on the page). 

Figure 4:
This figure shows a view of the numericallycomputed event horizon in a single slice, together with the locus of the event horizon’s generators that have not yet joined the event horizon in this slice, for a headon binary black hole collision. Notice how the event horizon is nondifferentiable at the cusp where the new generators join it. Figure reprinted with permission from [103]. © 1996 by the American Physical Society. 

Figure 5:
This figure shows a perspective view of the numericallycomputed event horizon, together with some of its generators, for the headon binary black hole collision discussed in detail by Matzner et al. [108]. Figure courtesy of Edward Seidel. 

Figure 6:
This figure shows the crosssection of the numericallycomputed event horizon in each of five different slices, for the headon collision of two extremal Kastor–Traschen black holes. Figure reprinted with permission from [46]. © 2003 by the American Physical Society. 

Figure 7:
This figure shows two views of the numericallycomputed event horizon’s crosssection in the orbital plane for a spiraling binary black hole collision. The two orbitalplane dimensions are shown horizontally; time runs upwards. The initial data was constructed to have an approximate helical Killing vector, corresponding to black holes in approximately circular orbits (the case of Grandclément et al. [78]), with a proper separation of the apparent horizons of . (The growth of the individual event horizons by roughly a factor of 3 in the first part of the evolution is an artifact of the coordinate choice – the black holes are actually in a quasiequilibrium state.) Figure courtesy of Peter Diener. 

Figure 8:
This figure shows the polar and equatorial radii of the event horizon (solid lines) and apparent horizon (dashed lines) in a numerical simulation of the collapse of a rapidly rotating neutron star to form a Kerr black hole. The dotted line shows the equatorial radius of the stellar surface. These results are from the D4 simulation of Baiotti et al. [21]. Notice how the event horizon grows from zero size while the apparent horizon first appears at a finite size and grows in a spacelike manner. Notice also that both surfaces are flattened due to the rotation. Figure reprinted with permission from [21]. © 2005 by the American Physical Society. 

Figure 9:
This figure shows the apparent horizons (actually MOTSs) for a spherically symmetric numerical evolution of a black hole accreting a narrow shell of scalar field, the 800.pqw1 evolution of Thornburg [155]. Part (a) of this figure shows (here labelled ) for a set of equallyspaced times between and , while Part (b) shows the corresponding MOTS radius and the Misner–Sharp [111], [112, Box 23.1] mass internal to each MOTS. Notice how two new MOTSs appear when the local minimum in touches the line, and two existing MOTS disappear when the local maximum in touches the line. 

Figure 10:
This figure shows the numericallycomputed apparent horizons (actually MOTSs) at two times in a headon binary black hole collision. The black holes are colliding along the axis. Figure reprinted with permission from [156]. © 2004 by IOP Publishing Ltd. 

Figure 11:
This figure shows the irreducible masses () of individual and common apparent horizons in a binary black hole collision, as calculated by two different apparent horizon finders in the Cactus toolkit, AHFinder and AHFinderDirect. (AHFinderDirect was also run in simulations at several different resolutions.) Notice that when both apparent horizon finders are run in the same simulation (resolution ), there are only small differences between their results. Figure reprinted with permission from [5]. © 2005 by the American Physical Society. 

Figure 12:
This figure shows the expansion (left scale), and the “generalized expansions” (left scale) and (right scale), for various surfaces in an Eddington–Finkelstein slice of Schwarzschild spacetime. Notice that all three functions have zeros at the horizon , and that while has a maximum at , both and increase monotonically with . 

Figure 13:
This figure shows the expansion for various pretracking surfaces, i.e. for various choices for the shape function , in a headon binary black hole collision. Notice how the three shape functions (34) (here labelled , , and ) result in pretracking surfaces whose expansions converge smoothly to zero just when the apparent horizon appears (at about ). Notice also that these three expansions have all converged to each other somewhat before the common apparent horizon appears. Figure reprinted with permission from [135]. © 2005 by the American Physical Society. 

Figure 14:
This figure shows the pretracking surfaces at various times in a spiraling binary black hole collision, projected into the black holes’ orbital plane. (The apparent slow drift of the black holes in a clockwise direction is an artifact of the corotating coordinate system; the black holes are actually orbiting much faster, in a counterclockwise direction.) Notice how, even well before the common apparent horizon first appears (, bottom right plot), the pretracking surface is already a reasonable approximation to the eventual common apparent horizon’s shape. Figure reprinted with permission from [135]. © 2005 by the American Physical Society. 
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